In some neighborhood in the f-plane of any ordinary point of the graph, the function f is a single-valued, continuous function.

The graph of f has at least one component whose support is the entire interval Aj.

" f one regards the Modest Proposal simply as a criticism of condition, about all one can say is that conditions were bad and that Swift's irony brilliantly underscored this fact ".

A choice function is a function f, defined on a collection X of nonempty sets, such that for every set s in X, f ( s ) is an element of s. With this concept, the axiom can be stated:

: For any set A there is a function f such that for any non-empty subset B of A, f ( B ) lies in B.

: There is a set A such that for all functions f ( on the set of non-empty subsets of A ), there is a B such that f ( B ) does not lie in B.

* There exists a model of ZF ¬ C in which there is a function f from the real numbers to the real numbers such that f is not continuous at a, but f is sequentially continuous at a, i. e., for any sequence

There is one final criterion that Hume thinks gives us warrant to doubt any given testimony, and that is f ) if the propositions being communicated are miraculous.

* An example of a non-unitary associative algebra is given by the set of all functions f: R → R whose limit as x nears infinity is zero.

* automorphism if f is both an endomorphism and an isomorphism.

* if f is an isomorphism in C, then F ( f ) is an isomorphism in D.

By setting K = ker ( f ) we immediately get the first isomorphism theorem.

In graph theory, an isomorphism between two graphs G and H is a bijective map f from the vertices of G to the vertices of H that preserves the " edge structure " in the sense that there is an edge from vertex u to vertex v in G if and only if there is an edge from ƒ ( u ) to ƒ ( v ) in H. See graph isomorphism.

* Let the index set I of an inverse system ( X < sub > i </ sub >, f < sub > ij </ sub >) have a greatest element m. Then the natural projection π < sub > m </ sub >: X → X < sub > m </ sub > is an isomorphism.

If M is simple, then f is either the zero homomorphism or injective because the kernel of f is a submodule of M. If N is simple, then f is either the zero homomorphism or surjective because the image of f is a submodule of N. If M = N, then f is an endomorphism of M, and if M is simple, then the prior two statements imply that f is either the zero homomorphism or an isomorphism.

A morphism f: Y → X is a limit of the diagram X if and only if f is an isomorphism.

Thus, one way to show two objects of C are distinct ( up to isomorphism ) is to show that their images under f are distinct ( i. e. not isomorphic ).

It follows that the biproduct A < sub > 1 </ sub > ⊕ A < sub > 2 </ sub > exists if and only if f is an isomorphism.

where the morphism on the left is the coimage, the morphism on the right is the image, and the morphism in the middle ( called the parallel of f ) is an isomorphism.

In fact, the parallel of f is an isomorphism for every morphism f if and only if the pre-abelian category is an abelian category.

It states that for the following commutative diagram ( in any abelian category, or in the category of groups ), if the rows are short exact sequences, and if g and h are isomorphisms, then f is an isomorphism as well.

The essence of the lemma can be summarized as follows: if you have a homomorphism f from an object B to an object B ′, and this homomorphism induces an isomorphism from a subobject A of B to a subobject A ′ of B ′ and also an isomorphism from the factor object B / A to B ′/ A ′', then f itself is an isomorphism.

It will be noted that point f has seven nearest neighbors, h and e have six, and p has only one, while the remaining points have intermediate numbers.

In any social system in which communications have an importance comparable with that of production and other human factors, a point like f in Figure 2 would ( other things being equal ) be the dwelling place for the community leader, while e and h would house the next most important citizens.

: For any set X of nonempty sets, there exists a choice function f defined on X.

Exports: $ 1. 225 billion f. o. b. ( 2008 )

Imports: $ 3. 546 billion f. o. b. ( 2008 )

The simple names s orbital, p orbital, d orbital and f orbital refer to orbitals with angular momentum quantum number l = 0, 1, 2 and 3 respectively.

The repeating periodicity of the blocks of 2, 6, 10, and 14 elements within sections of the periodic table arises naturally from the total number of electrons which occupy a complete set of s, p, d and f atomic orbitals, respectively.

** Haliotis brazieri f. hargravesi ( Cox, 1869 ) – synonym: Haliotis ethologus, the Mimic abalone, Haliotis hargravesi, the Hargraves ’ s abalone

In mathematics, an automorphism is an isomorphism from a mathematical object to itself.

In category theory, an automorphism is an endomorphism ( i. e. a morphism from an object to itself ) which is also an isomorphism ( in the categorical sense of the word ).

An isomorphism is simply a bijective homomorphism.

* Inverses: by definition every isomorphism has an inverse which is also an isomorphism, and since the inverse is also an endomorphism of the same object it is an automorphism.

* A group automorphism is a group isomorphism from a group to itself.

In this example it is not sufficient for a morphism to be bijective to be an isomorphism.

In a category with exponentials, using the isomorphism ( in computer science, this is called currying ), the Ackermann function may be defined via primitive recursion over higher-order functionals as follows:

Using Zorn's lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K.

It is essentially unique ( up to isomorphism ).

The isomorphism is defined by where for all we have

In fact, when A is a commutative unital C *- algebra, the Gelfand representation is then an isometric *- isomorphism between A and C ( Δ ( A )).

Sometimes two quite different constructions yield " the same " result ; this is expressed by a natural isomorphism between the two functors.

The two functors F and G are called naturally isomorphic if there exists a natural transformation from F to G such that η < sub > X </ sub > is an isomorphism for every object X in C.

Bicategories are a weaker notion of 2-dimensional categories in which the composition of morphisms is not strictly associative, but only associative " up to " an isomorphism.

In category theory, currying can be found in the universal property of an exponential object, which gives rise to the following adjunction in cartesian closed categories: There is a natural isomorphism between the morphisms from a binary product and the morphisms to an exponential object.

One can show that this map is an isomorphism, establishing the equivalence of the two definitions.